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Recent Progress on the Donaldson–Thomas Theory

Wall-Crossing and Refined Invariants

  • Book
  • © 2021

Overview

Authors:
  1. Yukinobu Toda

    1. Kavli IPMU, University of Tokyo, Chiba, Kashiwa-shi, Japan

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  • Provides an introduction of DT theory for both mathematicians and physicists
  • Emphasizes both the foundation and computations in the study of DT theory
  • Contains a mathematical theory of Gopakumar–Vafa invariants, a new subject not available in other survey works

Part of the book series: SpringerBriefs in Mathematical Physics (BRIEFSMAPHY, volume 43)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-viii
    Download chapter PDF

  2. Donaldson–Thomas Invariants on Calabi–Yau 3-Folds

    • Yukinobu Toda
    Pages 1-16

  3. Generalized Donaldson–Thomas Invariants

    • Yukinobu Toda
    Pages 17-28

  4. Donaldson–Thomas Invariants for Quivers with Super-Potentials

    • Yukinobu Toda
    Pages 29-38

  5. Donaldson–Thomas Invariants for Bridgeland Semistable Objects

    • Yukinobu Toda
    Pages 39-51

  6. Wall-Crossing Formulas of Donaldson–Thomas Invariants

    • Yukinobu Toda
    Pages 53-69

  7. Cohomological Donaldson-Thomas Invariants

    • Yukinobu Toda
    Pages 71-82

  8. Gopakumar–Vafa Invariants

    • Yukinobu Toda
    Pages 83-90

  9. Some Future Directions

    • Yukinobu Toda
    Pages 91-94

  10. Back Matter

    Pages 95-105
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Keywords

About this book

This book is an exposition of recent progress on the Donaldson–Thomas (DT) theory. The DT invariant was introduced by R. Thomas in 1998 as a virtual counting of stable coherent sheaves on Calabi–Yau 3-folds. Later, it turned out that the DT invariants have many interesting properties and appear in several contexts such as the Gromov–Witten/Donaldson–Thomas conjecture on curve-counting theories, wall-crossing in derived categories with respect to Bridgeland stability conditions, BPS state counting in string theory, and others. 

Recently, a deeper structure of the moduli spaces of coherent sheaves on Calabi–Yau 3-folds was found through derived algebraic geometry. These moduli spaces admit shifted symplectic structures and the associated d-critical structures, which lead to refined versions of DT invariants such as cohomological DT invariants. The idea of cohomological DT invariants led to a mathematical definition of the Gopakumar–Vafa invariant, which was firstproposed by Gopakumar–Vafa in 1998, but its precise mathematical definition has not been available until recently.


This book surveys the recent progress on DT invariants and related topics, with a focus on applications to curve-counting theories.


Reviews

“The book is directed at readers with a solid foundation in algebraic geometry. … the main definitions and theorems are nicely illustrated by examples. … The book will serve as a guide to further reading for those wishing to learn more details about the theory.” (Matthew B. Young, Mathematical Reviews, March, 2023)

Authors and Affiliations

  • Kavli IPMU, University of Tokyo, Chiba, Kashiwa-shi, Japan

    Yukinobu Toda

About the author

The author is currently Professor and Principal investigator at Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo. He was an invited speaker at the ICM 2014.

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